91Ó°ÊÓ

The rules for computing derivatives are fundamental in calculus. They help us find the rate at which a function is changing at any given point. Here are some key rules to consider:

• Constant Rule: The derivative of a constant is always zero.
• Power Rule: If you have a function in the form of \(f(x) = x^n\), its derivative is \(f'(x) = nx^{n-1}\).
• Sum Rule: For two functions \(f(x)\) and \(g(x)\), the derivative of their sum is the sum of their derivatives \( (f(x) + g(x))' = f'(x) + g'(x)\).
• Constant Multiple Rule: If you have a constant \(c\) and a function \(f(x)\), the derivative of their product is the product of the constant and the derivative of the function \(cf(x) = cf'(x)\).

In our exercise, we used the Sum Rule and the Constant Multiple Rule to derive \(h'(x)\) from \(h(x) = 3 f(x) + 2 g(x)\), thus getting \(h'(x) = 3 f'(x) + 2 g'(x)\). These rules greatly simplify the process of finding the derivative of a complex function.

Function evaluation involves finding the value of a function for a particular input. This is a key concept when working with functions in mathematics.

In our exercise, we evaluated the function \(h(x) = 3 f(x) + 2 g(x)\) at \(x = 5\). Here's how we did it:

1. We were given values for \(f(5)\) and \(g(5)\) as part of the problem.2. We then substituted these values into the equation for \(h(x)\).

This gave:\[ h(5) = 3 f(5) + 2 g(5) \] Using the values \(f(5) = 2\) and \(g(5) = 4\), we substituted them:\[ h(5) = 3(2) + 2(4) = 6 + 8 = 14 \]

Function evaluation helps to break down the steps and get precise values at specific points. Always remember to verify and substitute values correctly.

Combining functions is a process that allows us to create new functions by combining simpler functions using operations like addition, multiplication, etc.

In our example, we combined the functions \(f(x)\) and \(g(x)\) to form a new function \(h(x)\):\[ h(x) = 3 f(x) + 2 g(x) \]
Here, we used addition and multiplication to combine these functions. This is a linear combination because it only involves constant multiples and sums.

When working with combined functions, it’s essential to understand each individual function and how the operations affect them. This involves:

• Performing operations step-by-step, ensuring each function is correctly substituted.
• Applying derivative rules accurately when deriving combined functions.
• Evaluating each part of the combined function independently before combining them.

In our exercise, after creating \(h(x)\), we moved on to find its derivative. By applying the Sum Rule and Constant Multiple Rule, we derived \(h'(x)\), which provided:\[ h'(x) = 3 f'(x) + 2 g'(x) \]We then evaluated this at \(x = 5\) using the provided values for \(f'(5)\) and \(g'(5)\).Combining functions enables us to solve more complex problems by simplifying into manageable parts.